Problem: For all values of $x$ for which it is defined, $f(x) = \cot \frac{x}{4} - \cot x$ can be written as
\[f(x) = \frac{\sin kx}{\sin \frac{x}{4} \sin x}.\]Find the value of $k.$
We can write
\[\cot \frac{x}{4} - \cot x = \frac{\cos \frac{x}{4}}{\sin \frac{x}{4}} - \frac{\cos x}{\sin x} = \frac{\cos \frac{x}{4} \sin x - \sin \frac{x}{4} \cos x}{\sin \frac{x}{4} \sin x}.\]From the angle subtraction formula,
\begin{align*}
\frac{\cos \frac{x}{4} \sin x - \sin \frac{x}{4} \cos x}{\sin \frac{x}{4} \sin x} &= \frac{\sin (x - \frac{x}{4})}{\sin \frac{x}{4} \sin x} \\
&= \frac{\sin \frac{3x}{4}}{\sin \frac{x}{4} \sin x},
\end{align*}so $k = \boxed{\frac{3}{4}}.$